It may on the surface. Wittgenstein agrees with Mcdowell that the world as we perceive arrives already conceptually imbued, but while Mcdowell believes human thinking is rationally answerable to the world, Wittgenstein doesn’t place mind on one side and world on the other of such a divide. We don’t mirror or answer to a world, we enact a world via discursive practices.
But that’s the picture McDowell rejects too. A big part of the Mind and World project is to dissolve the idea of a borderline between mind and world, with experience located uneasily on one side or the other. As he says, a “dualism of scheme and Given” forces us to posit something outside “the space of reasons” (“spontaneity,” for McDowell) that can nonetheless provide warrant for our empirical beliefs. It is the dualism itself that McDowell argues against; he doesn’t thnk the “something” is outside the space of reasons at all. He believes that we are constrained by what is “outside the activity of thinking, though not from outside of what is thinkable, so not from outside the space of concepts.” (144)
In short, “The idea of a structure that must be found in any intelligible conceptual scheme need not involve picturing the scheme as one side of a scheme-world dualism.” (158). I 'm figuring Witt would agree, and McDowell certainly cites him in a number of places, though what I quoted to @Sam26 is the only direct reference to hinges, I believe.
McDowell rejects a scheme/content dualism. He doesn’t want a conceptual scheme on one side and a brute world on the other. Experience is already conceptually structured. The split in Mcdowell isnt between two kinds of substances, one mental and the other natural, but between the judging and the world being judged, that is, between thought and reality.
For McDowell, the central philosophical problem is how thought is answerable to reality without invoking a Given. For Wittgenstein, the notion of answerability itself is already embedded in our practices of judging, correcting, teaching, and responding to the world.
McDowell wants to explain how empirical judgment is constrained by reality. Wittgenstein dissolves the demand for such an explanation by reminding us that our ordinary practices already constitute what counts as being constrained by reality.
Yes, that’s largely how I read McDowell too, though I think even a “split between thought and reality” takes him too far towards the duality he’s rejecting. Glad you agree that the issue is not “mind on one side and world on the other of such a divide.”
I wrote a paper about a year ago on what I saw as a connection between Wittgenstein’s hinges and Godel’s incompleteness theorems. I recently rewrote the paper with a more appropriate title: Wittgenstein’s Hinges and Godel’s Incompleteness Theorems: An Analogous Limit to Internal Justification
I tried using more precise language to show the connection. I’ll post the paper for your consideration. To my knowledge no one has made this connection.
Post 1
Abstract
In Ludwig Wittgenstein’s final notes, published posthumously as On Certainty (1969), Wittgenstein describes what stands fast for us, viz., what lies beyond justification and doubt (OC 341-343): what philosophers and others have called hinges, or hinge-propositions. They support our language-games and epistemic practices, and they offer a picture of knowledge that challenges traditional epistemology’s criteria that everything we believe be justified. In this paper I argue for a structural parallel between Wittgenstein’s hinges and Gödel’s 1931 incompleteness theorems. Both thinkers, I believe, uncover an analogous limit to internal justification: Wittgenstein shows that our epistemic practices rest on hinges embedded in our form of life that the practices themselves cannot justify, while Gödel shows that any consistent, effectively axiomatized system strong enough for arithmetic rests on axioms and the presumption of consistency it cannot establish from within, and contains arithmetical sentences it can neither prove nor refute. These ungrounded foundations are not failures of reasoning as some might think; they are necessary conditions that make systematic inquiry possible. The parallel suggests that ungrounded foundations enable rather than undermine knowledge, a structural feature shared by these two domains. This has implications for how we think about certainty in both epistemology and the philosophy of mathematics.
A good metaphor is ‘The Matrix: Codes in the Walls’. There’s room for something, but what is that something is the question.
I have narrowed it down to brain-space, body-matter and offline or online mind. This leaves what we interact with, ‘the codes in Walls’, I have aptly named Lightframe.
We’re meant to be doing something during this seemingly serene time.
Post 2
Introduction
We perform countless actions without hesitating for a moment. We sit on chairs, pick up pencils, and walk across rooms without ever questioning the existence of the chair, the pencil, or the floor beneath our feet. This unthinking confidence illustrates what Wittgenstein describes using the hinge metaphor: something that stands fast for us, supporting our use of language and our epistemic practices. Wittgenstein compares them to the hinges on which a door turns, namely, “If I want the door to turn, the hinges must stay put” (OC 343). The door swings freely, the questioning and answering go on, precisely because the hinges remain fixed.
I will argue that Wittgenstein’s hinges bear a structural resemblance to Gödel’s incompleteness theorems of 1931. I am not claiming that the analogy is exact, nor that Wittgenstein and Gödel were engaged in the same project; they obviously were not. What I am claiming is that both thinkers, working in very different domains, uncovered an analogous structural limit, namely, that a system of inquiry depends on certain enabling conditions that are not themselves established by its own internal procedures. It is this connection that I will examine.
Traditional epistemology often misreads hinges by forcing them into the role of ordinary propositions, that is, statements to be judged true or false, justified or unjustified. But this neglects their peculiar role in our epistemic form of life (OC 136-138). Hinges are not conclusions of our epistemic practices; they are the bedrock those practices stand on. They come before argument and evidence, not after. Similarly, Gödel’s incompleteness theorems show that any consistent, effectively axiomatized system strong enough for arithmetic contains arithmetical sentences that can be neither proved nor refuted within the system, and that no system, if it is consistent, can prove its own consistency from within.
Why does this connection matter? It matters because it marks the boundary of what counts as bedrock for both epistemic and mathematical systems. Both depend on starting points that lie beyond justification, starting points that are not flaws in our reasoning but necessary foundations that make knowledge claims possible. This paper argues that ungrounded foundations enable knowledge rather than undermine it, and that hinges and the unproven foundations of a formal system serve a similar purpose. The hinge is not a defect in the door; it is what lets the door turn.
If we’d all associate ourselves with such a thesis, we’d accept, for example, that pain exists, as with the ‘does pain exist?’ thread.
What’s the level of ‘whatness’ as outlined by the theory, or ‘hereness’?
What’s a door with hinges?
Post 3
Section 1: Hinges and Their Foundational Role
The hinge metaphor is crucial to Wittgenstein’s final notes. In On Certainty, Wittgenstein describes what stands fast, grounding our epistemic language-games. Wittgenstein never explicitly sorts hinges into types, but his examples suggest a distinction between nonlinguistic and linguistic varieties, that is, different levels of fundamental certainty.
Nonlinguistic hinges are the most basic, they are the bedrock certainties that ground our actions and our interactions with the world. These are not expressed as propositions to be justified or doubted; they are embodied in our unreflective action. For example, the certainty that the ground will support us when we walk. Do we test the floor each morning before stepping out of bed? Do we make a study of chairs before sitting down? Of course not, we simply walk, and we simply sit. Our confidence that objects behave in predictable ways (that chairs hold our weight, that pencils mark paper) operates beneath the level of articulation, forming the silent background against which all conscious thought and language become possible.
Many hinges can be put into words without ceasing to be fundamental. When I say “I have two hands” or “the earth exists,” I give linguistic expression to a certainty already shown in how I act; the sentence describes the hinge, but the hinge was operating long before it and does not depend on the statement for its standing. Statements such as “I am a human being” or “the world has existed for a long time” look like ordinary propositions, and they have, so to speak, the look and smell of normal propositions, but their use is altogether different. They function as structural supports for our discourse rather than as claims requiring a justification. These verbalized hinges remain world-grounded. They answer to a world that constrains our practices, and putting them into language does not change their basic status.
Other hinges, however, exist only within a language-game and have no footing outside it, and this distinction marks two ends of a range, with the clearest cases at each end. That bishops move diagonally is a fundamental certainty for anyone playing chess, but there is no pre-linguistic, embodied fact beneath it; the rule exists only because the practice of chess exists. If we set worldly engagement aside, “the earth exists” loses the context in which it does work; set the practice of chess aside and “bishops move diagonally” has nothing left to mean. It is in this sense that some linguistic hinges are less fundamental than nonlinguistic ones, not because they are spoken, but because they are constituted by a contingent human practice, whereas a hinge like our certainty that the ground will hold answers to the world. Chess is the clearest case of such a practice-internal hinge. Not every hinge sorts so cleanly: “the world has existed for a long time” both answers to a world and is embedded in our language-soaked practices of memory and history, sitting somewhere between the two extremes.
These distinctions show that certainty operates at different depths in grounding knowledge. Nonlinguistic, world-grounded hinges form the deepest stratum, the unquestioned backdrop that makes any questioning or justification possible, whether or not we ever put them into words. Purely practice-internal hinges, such as the rules of a game, are foundational within their practice but rest on nothing deeper than that practice. Both resist justification, but for different reasons: the world-grounded ones because they are enacted prior to reason, the practice-internal ones because they are the constitutive rules within which reasons are given at all.
Here Wittgenstein breaks with traditional epistemology. Rather than treating these certainties as beliefs in need of justification, he recognizes them as the ungrounded ground that makes justification itself possible. He writes, “Why do I not satisfy myself that I have two feet, when I want to get up from my chair? There is no why. I simply don’t. This is how I act” (OC 148). There is no why. To doubt these hinges would collapse the very framework within which doubt makes sense; it would be like sawing off the branch on which one sits.
An important distinction must be drawn between the subjective and objective dimensions of these kinds of certainties. Our relationship to hinges is one of unquestioning acceptance, but this certainty is not merely psychological, that is, it is not simply a feeling of conviction. It is objective in the sense that it is not a private feeling we could decide to adopt or drop, that is, it is shaped and held fast by a world we share and act within. As Wittgenstein puts it, our picture of the world is “the inherited background against which I distinguish between true and false” (OC 94).
This interpretation of hinges as operating at different foundational levels finds support in recent Wittgenstein scholarship, though it diverges from some prominent interpretations. Danièle Moyal-Sharrock argues that hinges are fundamentally non-propositional, that is, they exist as lived certainties rather than as beliefs or knowledge claims (Moyal-Sharrock 2004). My account of the world-grounded hinges aligns with her emphasis on their embodied, pre-propositional character. However, I suggest that some hinges, the practice-internal ones, function at a more articulated level within our language-games, even though they too resist the usual patterns of justification.
Duncan Pritchard emphasizes that hinges are commitment-constituting rather than knowledge-constituting, that is, they form a distinct epistemic category that enables knowledge without being knowledge (Pritchard 2016). This view supports the parallel with mathematical axioms. In other words, both hinges and axioms function as enabling commitments that make systematic inquiry possible without themselves being objects of that inquiry. The mathematical case strengthens Pritchard’s insight by showing that even formal domains require such commitment-constituting foundations.
This analysis extends beyond epistemology. It reveals a striking parallel with Gödel’s incompleteness theorems, which demonstrate analogous limits within formal mathematical systems. Just as Gödel showed that sufficiently strong systems contain sentences they cannot settle and cannot establish their own consistency from within, Wittgenstein’s hinges reveal that our epistemic systems rest on certainties that cannot be justified internally. The comparison suggests a structural limitation in rational discourse, whether in mathematics or in everyday knowledge, and it invites us to reconsider what it means for knowledge to be grounded.
Post 4
Section 2: Gödel’s Incompleteness Theorems and a Hinge-Like Limit
Gödel’s incompleteness theorems of 1931 mark the hard limits within formal theories. In any consistent, axiomatized system strong enough for arithmetic, there are arithmetical sentences that can be neither proved nor disproved within the system. From a view outside the system, such a sentence can be recognized as true under the standard interpretation of arithmetic, but that truth is not established by the system’s own proof mechanisms; there is a limit that is always relative to any system. The second theorem adds that no system, if it is consistent, can prove its own consistency from within, on the standard formalization of that claim. These are structural limits, not defects of a particular set of axioms, and they cannot be removed by adding more axioms. If the system is extended with new axioms to settle an undecidable sentence, and as long as the strengthened theory remains consistent, that is, axiomatized, and remains strong enough for arithmetic, new undecidable sentences will naturally arise.
Independently of Gödel, formal theories begin with axioms that are adopted rather than proved. For example, think of the rules of chess. No one proves that bishops move diagonally. The rule is not a conclusion reached by playing; it is what makes playing possible. One does not justify the rules of chess from within chess, that is, the rules, the board, and the pieces are the backdrop that gives life to the game. Mathematical axioms function in the same way, that is, they are the starting points one adopts in order to prove anything. What holds up the axioms? Nothing holds up the axioms, which is why they are axioms. We have reached the bottom, or rather, the beginning.
Gödel’s results then add another limit, and this is where his contribution lies. Even once the axioms are fixed, some truths remain unprovable, and the theory cannot certify its own consistency from within. This second point deserves emphasis. Every proof carried out in a formal system leans on the system’s consistency, because an inconsistent classical system proves every statement, so that proof within it loses the power to discern truth from falsehood. Consistency is presupposed in every step the mathematician takes; yet it is precisely what the system cannot establish. The mathematician’s confidence in the system, like our confidence that the floor will hold, is not the result of a proof internal to the practice; it is one of the things the practice necessarily depends on. Here the resemblance to hinges is at its closest, namely, a certainty presupposed by the whole practice, relied upon at every move, and not securable by the practice itself.
This limitation mirrors Wittgenstein’s hinges in an important way, but it is worth being precise about what plays the hinge-like role. It is not the undecidable sentence, which reveals the limit rather than serves as a foundation; it is the axioms, the rules of inference, and the presumption of consistency on which the system runs but which it cannot certify from within. Just as hinges are certainties that are not justified by the practice they enable, so to Gödel identifies a limit on internal vindication even very strong formal systems cannot establish their own consistency from within. The point is not that axioms ought to be proved (axioms are adopted) but that every practice, including mathematics, runs on enabling commitments that do not receive their warrant from the conclusions they make possible.
There is an important difference here, and it should be explained. Mathematical axioms are in many instances chosen for their elegance, consistency, and power to generate interesting mathematics, while hinges are embedded in the contingent cultural and biological circumstances of human life. Does this difference weaken the parallel I’ve suggested? I would argue the opposite. If even mathematics, the very paradigm of rigorous proof, requires foundational starting points it cannot justify from within, how much more must everyday understanding rely on unexamined certainties? That two domains as different as formal mathematics and lived experience share this requirement is striking, even if it stops short of some universal law about how all systems must be organized.
Both domains thus reveal that functioning without such foundational elements is improbable. Mathematical systems cannot get off the ground without axiomatic starting points, just as our epistemic practices would collapse without the bedrock certainties Wittgenstein identifies. Try to imagine chess without the board and pieces; one wonders if the game would even get off the ground. The parallel illuminates a shared structural necessity, that is, in both domains, inquiry requires ungrounded foundations, and these foundations enable rather than undermine the reasoning that takes place within the system.
It should be acknowledged that Wittgenstein himself was famously dismissive of Gödel’s theorems in his Remarks on the Foundations of Mathematics, and I am not claiming that he would have endorsed this comparison. My claim concerns the structure that On Certainty uncovers, namely, that our practices rest on foundations they cannot justify from within, a structure that Gödel’s results point out in mathematics, whatever Wittgenstein thought of Gödel’s proof.
Post 5
Section 3: Beyond Internal Justification: A Cross-Domain Analysis
Both Wittgenstein and Gödel reveal that justification operates within boundaries. In each domain, certain elements of the domain serve as foundations that cannot be justified within the system they support. Both thinkers expose analogous structural features, namely, the impossibility, in either domain, of a complete system of justification.
Traditional approaches to knowledge often assume that justification requires tracing our claims back to bedrock, which are themselves justified. But this assumption generates the classical problem of infinite regress. If every foundation requires a further justification, then the chain of justification is never secure; we are left holding up the foundation with another foundation, and that one with another, and so on. One does not support 10x10s with 2x4s, and one cannot support a foundation with what stands on it. Both Wittgenstein’s hinges and Gödel’s incompleteness results reveal why the demand for complete internal justification is not merely a problem that cannot be satisfied, but impossible in principle.
As Wittgenstein points out, “There is no why. I simply don’t. This is how I act” (OC 148). This captures something crucial about hinges, namely, they are arational, that is, they are prior to rational evaluation, enabling rational discourse rather than emerging from it. Hinges are not conclusions we reach through any method of justification; they are lived realities that make justification possible. Similarly, mathematical axioms are not theorems we prove; they are starting points we adopt in order to make proof possible. Justification comes to an end, and the end is bedrock.
Again, an important difference emerges between the domains, and Wittgenstein himself points us it out. Axioms are presuppositions, that is, statements adopted and laid down at the start of a theory; they are selected through systematic processes, for their mathematical power and elegance. Hinges run deeper. “The end is not an ungrounded presupposition: it is an ungrounded way of acting” (OC 110). A hinge is not something we lay down; it is something we live. Hinges reflect the biological and cultural states of affairs of our existence, and they show themselves in what we do, not in what we postulate. If anything, this difference makes the parallel more compelling by displaying its scope, that is, if even the most rigorous formal disciplines require unjustified starting points, the necessity of such foundations in everyday knowledge becomes all the more apparent.
This cross-domain similarity reveals a structural feature shared by both domains. Neither formal mathematical theories nor epistemic frameworks can achieve complete self-justification. Each requires elements that are not justified within the system but that make inquiry within the system possible. And these unjustified foundations are not failures or limitations; they are enabling conditions, that is, they are what make coherent thought and practice possible at all.
Recognizing this transforms how we understand the relationship between certainty and knowledge. Instead of treating unjustified elements as epistemological problems crying out for solutions, we can understand them as necessities that allow knowledge systems to function. Both mathematical proof and everyday understanding depend on foundations that lie beyond their internal capacity for justification, and yet it is this very dependence that enables their respective forms of inquiry. The door turns because the hinges stay put.
Post 6
Conclusion
I have argued for a structural parallel between Wittgenstein’s hinges and Gödel’s incompleteness results: each demonstrates that any system of inquiry rests on ungrounded foundations; and by examining how epistemic and mathematical systems share this feature, we gain insight into the ungrounded foundations on which these two domains rest.
The parallel between these seemingly distant insights suggests that the limits of internal justification are not accidental features of particular systems but, at least in these two domains, necessary conditions for the inquiry they make possible. Recognizing this gives us a more complete picture of how epistemology functions. Specifically, not through endless chains of justification reaching some ultimate self-certifying ground, but through practices and formal systems that rest on foundations lying beyond their internal structure.
Rather than treating these limits as problems which need solving, we should take them as structural conditions of inquiry. Wittgenstein’s hinges anchor our epistemic practices in the lived background of a form of life; in mathematics, axiomatic choices provide the starting points of a theory. Gödel’s incompleteness results mark the corresponding boundary on internal vindication, since even with the axioms fixed, a system strong enough for arithmetic has sentences it cannot settle and cannot, from within, prove its own consistency. Both lessons point in the same direction, namely, that the demand for a completely self-grounding system is not merely difficult to meet but misconceived.
I believe this perspective has broader implications for how we understand certainty and knowledge. The interplay between grounded and ungrounded elements is not a flaw in human reasoning but a necessary feature of inquiry in both domains. Whether the same structure appears in other domains would require separate argument, but the comparison developed here gives us reason to take that possibility seriously.
References
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173-198.
Moyal-Sharrock, D. (2004). Understanding Wittgenstein’s On Certainty. Palgrave Macmillan.
Pritchard, D. (2016). Epistemic Angst: Radical Skepticism and the Groundlessness of Our Believing. Princeton University Press.
Wittgenstein, L. (1969). On Certainty (G. E. M. Anscombe & G. H. von Wright, Eds.; D. Paul & G. E. M. Anscombe, Trans.). Basil Blackwell.
Wittgenstein, L. (1978). Remarks on the Foundations of Mathematics (G. H. von Wright, R. Rhees, & G. E. M. Anscombe, Eds.; G. E. M. Anscombe, Trans.; Rev. ed.). Basil Blackwell.